1) Can you post a screen shot -- the only time I've ever been asked for an Office product key is when you start the INSTALL process. After installation you will get an Activate Office screen the first time you run any office application.
2) what Office programs open ?. you should only basically have Powerpoint, EXCEL and WORD applications in the student and teacher office editions.
3) Although this should really be 1) - Did you INSTALL office 2010 correctly. It's possible also you had some other versions of office on the machine too --these should really be removed before installing Office 2010.